Fast structured AMG preconditioning for the bidomain model in electrocardiology

The electrical activity of the heart may be modeled by a nonlinear system of partial differential equations known as the bidomain model. Due to the rapid variations in the electrical field, accurate simulations require a fine-scale discretization of the equations and consequently the solution of large severely ill-conditioned linear systems at each time step. Solving these systems is a major bottleneck of the whole simulation. We propose a highly effective preconditioning strategy for a general and popular three-dimensional formulation of the problem. A theoretical analysis of the preconditioned matrix ensuring mesh independence of the spectrum is also described. Numerical comparisons with state-of-the-art approaches confirm the effectiveness of our preconditioning technique. Finally, we show that an equivalent but less exercised formulation provides the best performance, in terms of CPU time.